Welcome! I hope you'll find the advice you're looking for. Admittedly, I am no expert on these particular games, but from I understand, this generation of Data East games are nearly identical to the hardware used in William's System 11 games (except for a few hardware differences here and there).

So, for the flippers, start easy. Check all the fuses. The flippers should be fused in the backbox on the PPB board. If the fuses check out, check for 50 volts at the flipper coil. Do this part with a game started. Also, while you're under there, check the end-of-stroke switches. They should be normally closed.

If that all checks out, there's a relay that engages the flippers. It's enabled from a transistor on the CPU board. Looking in the manual, it's a little vague, but I think It's Q80--don't quote me on that though.

Same goes for your pop bumper. Honestly, if it smelled like it was burning, it might be too late. Check for continuity though. Anything less than 2 ohms is a short. Otherwise, I'll bet that it's another fried transistor that's causing it to lock on. According to the manual, that'd be either Q8, Q9, or Q11. Those are the transistors for the pops. Again, I'm not sure which transistor corresponds to the pop bumper you're having issues with.

Testing procedures can be found on pinwiki. Not sure if you've been there yet, but it's a fantastic resource. There's also a ton of guys here that are far more useful than I am, but I at least hope I can steer you in the right direction. Good luck!

That’s a big red flag. Something is causing it to blow. With a multimeter, check your flipper coils for any shorts with the continuity setting. Also check the end of stroke switch to make sure it’s opening with a full flipper extension.

There's definitely a dead short someplace, I'm guessing. It might be one of the bridge rectifiers.

Oh man... Yikes. Could be much worse, but now you have some board work ahead of you. Here's what you do moving forward: Unplug the game, check every single fuse, and compare it against what is specified in the manual. Replace any over/under fused ones with proper replacements. Check all the bridge rectifiers, Then, under the playfield, check the ohms of every single coil. Each one. Admittedly, that's usually a good baseline strategy for any new-to-you game. But check each one. Anything under 2 ohms is a dead short, and should be replaced. While you're at it, check the diodes across the lugs of each coil. They're typically 1N4004 diodes, and their primary function is to prevent an EMF spike from taking out components on the CPU/Driver board. Replace any that are broken or open or shorted. Same goes for the flipper coils. When you replace them, the banded side of the diode is wired into the power side of the wires going into the coil. Power cables are almost always noted by the thicker wire, or sometimes there are two wires daisy chained.

But familiarize yourself with the way the game works by reading up on all the material you can from pinwiki. That'll give you a good idea of what you'll be up against, and give you good testing procedures for diodes, bridges, and switches.

I wish I could tell you. I've never owned a Data East game, so my knowledge about them stops once it regards something that's not congruent with William's boardset, and the PPB board was proprietary DE stuff. If I were a betting man, I'd say that it was similar in function to William's Aux Driver Board, and if that's the case, then only components needed for the particular game are populated, and that one transistor that's not the same as the others, has been replaced with a different kind over the years (probably in an effort to fix the same thing you're working on).

This is pretty much the extent of my knowledge though. I do know that DataEast (much like William's) used what was called "switched" solenoids for things like their pop bumpers and sling shots. These were only "half" computer controlled. That is, the computer would activate them in diagnostics for testing, but not in-game. In game, they would be directly controlled by a switch that would close a circuit and tell the solenoid to activate. This was done because they needed instantaneous response. If the switch is stuck closed, the solenoid will lock on (or machine gun). It may be wise to check the switches for solder splash shorts, and proper adjustment as well. Compare it to a pop bumper that appeared to be working. I'm almost certain this is how they work in Time Machine.